Questions related to various forms of integration including the Riemann integral, Lebesgue integral, Riemann–Stieltjes integral, double integrals, line integrals, contour integrals, surface integrals, integrals of differential forms, ...

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6
votes
1answer
209 views

Henstock, Differentiation under the integral sign

Does anyone know, where I can find the proof of necessary and sufficient conditions for differentiating under the integral sign in case of Henstock integral? Here are the theorems but not all the ...
14
votes
1answer
858 views

Calculate in closed form $\int_0^1 \int_0^1 \frac{dx\,dy}{1-xy(1-x)(1-y)}$

The following question has a 500 points bounty on MSE that soon comes to an end, and no answer as expected was given yet. How would a professional solve the problem? Wish you succcess. ...
2
votes
0answers
66 views

expectation involving normal pdf and Rayleigh distribution

I need to calculate following definite integral \begin{equation*} \frac{1}{2\pi }\int_0^\infty \frac{x^2 e^{-x^2/\sigma^2 } }{\sigma} \frac{e^{-\frac{\lambda}{{ax^2+b}}}}{\sqrt{ax^2+b}} ~~dx. ...
10
votes
2answers
359 views

Computing Gauss Legendre Quadrature for Large N

I've been scanning across the web, and haven't found a good method to compute the Gauss Legendre abscisas and weights $\{ x_j, w^j \} _{j=1}^N$ for large $N\in\mathbb{N}$. My question is how to do it, ...
2
votes
1answer
113 views

Solving complicated equation involving integral of error functions

I've been trying to solve the following equation for $\sigma$ $$\sigma^2 = \int_0^1 \left\{ \frac{1}{2} \operatorname{erfc} \left[ \frac{x + B}{\sqrt{2 (Rp - 1)} \, \sigma} \right] + \frac{1}{2} ...
0
votes
1answer
148 views

Is the following “section-wise” defined function measurable in the product space?

I asked this question in mathstackexchange a couple of days ago. Almost right after posing it a partial (affirmative) answer came to my mind in the following form Proposition: Assume that ...
4
votes
0answers
84 views

Numerical integration error bounds on the unit sphere

A sequence of points $x_1,x_2,\dots$ on the unit sphere $S^{D-1}$ is said to be uniformly distributed if \begin{align} \lim_{N \rightarrow \infty} \frac{1}{N} \sum_{j=1}^N f(x_j) = \int_{x \in ...
2
votes
1answer
515 views

Real and imaginary part of an holomorphic function

I guess this could be a very elementary question. Anyway I can not find an answer in literature. Let $f:U\rightarrow\mathbb{C}$ be an holomorphic function on an upen subset $U\subseteq\mathbb{C}$. ...
2
votes
2answers
325 views

Integrating a barycentric monomial over a simplex

Are there standard formulas for the integral over a simplex of a monomial in the barycentric coordinates? Can someone supply a reference? I think I have seen such formulas, but I am unable to find ...
1
vote
0answers
400 views

What does the Riemann–Stieltjes integral measure? [closed]

The Riemann–Stieltjes integral is a generalization of the Riemann integral, and has a definition based on a sum analogous to the Riemann sum: $$ S(P,f,g) =\sum_{k=1}^{n} f(x_k)\Delta g(x_k) $$ where ...
11
votes
0answers
250 views

Reference for a proof of the fiberwise Stokes theorem

The fiberwise Stokes theorem says that given a differential form on a smooth fiber bundle whose fibers have boundary, the difference between the fiberwise integral of the differential and the ...
5
votes
1answer
145 views

Asymptotics of Fresnel integrals

It is known that \begin{equation*} I(p) = \sqrt { \frac {4 \mathrm{i} p} {\pi}}\int \limits _{-\infty} ^{\infty} \mathrm{e}^{- \mathrm{i} p x^2} \varphi (x) \mathrm{d}x \end{equation*} is a bounded ...
2
votes
1answer
149 views

acoustic dipole volume integral w/ dirac delta?

I have an acoustic research problem that leads to the following integral formulation: \begin{align} \int_{-\infty}^{\infty}\int_{-\infty}^{\infty}p(\mathbf{y},\tau)\frac{\partial}{\partial ...
6
votes
0answers
106 views

Integral identity related with cubic analogue of arithmetic-geometric mean

This is re-post from MSE as I did not get an answer there. Let $a,b$ be positive real numbers and we define two sequences $\{a_{n}\},\{b_{n}\}$ as follows: ...
1
vote
0answers
104 views

What is $\int (1-e^{-x})^n dx$? [closed]

For my purposes, $n$ is a non-negative integer, and $x > 0$. I didn't know how to evaluate this integral, so I plugged it into Mathematica. It told me the solution is $(-1)^n B(e^x; -n, n+1)$ I ...
7
votes
2answers
467 views

Interesting triple integral

Some time ago I stumbled on an alleged identity $$\int\limits_0^\infty \frac{dx}{x} \int\limits_0^x \frac{dy}{y} \int\limits_0^y \frac{dz}{z} [\sin{x}+\sin{(x-y)}-\sin{(x-z)}-\sin{(x-y+z)}]= ...
2
votes
1answer
99 views

Asymptotic expansion of a Laplace-type integral with a “manifold of maxima”

Moved this here from MathSE because I had no luck there and suspect the question may be harder than I first thought. Consider the integral $$ I(\alpha)=\int_0^1 dx_1 \int_0^1 ...
2
votes
1answer
204 views

A conjecture regarding the integral of the square of an entire function

Can some help me prove or disprove the following assertion which I encountered in research? Thanks! Let $f:\mathbb R\to\mathbb R$ be an analytic function. If for $\forall c > 0$, we can find some ...
4
votes
1answer
454 views

Help with the integral $\int_{0}^{\infty}\log\left(1+\frac{s^{2}}{4\pi^{2}} \log^{2}(1+ix)\right ) e^{-2\pi nx}dx$

We have the integral : $$\int_{0}^{\infty}\log\left(1+\frac{s^{2}}{4\pi^{2}} \log^{2}(1+ix)\right ) e^{-2\pi nx}dx$$ Where s is a complex parameter, and n is a positive integer. The integral ...
4
votes
3answers
382 views

Area of metric spheres in Riemannian manifolds

I am trying to estimate the integral $\int \mathbb{e} ^{-d(x_0,x)^2} \mathbb{d}x$ on a Riemann manifold $(M,g)$, for some arbitrary fixed $x_0 \in M$ and $d$ the usual distance. The only thing that I ...
12
votes
1answer
446 views

Are there integral representations of the Mertens constant?

It is well-known that the Euler constant $$\gamma=\lim\limits_{n\to \infty}\biggl( \sum\limits_{k\le n}\dfrac{1}{k}-\ln{n}\biggr ) $$ has a bunch of integral representations, e.g. ...
4
votes
1answer
241 views

Motivic integration in positive characteristic: how much is known?

It seems that in papers on motivic integration people usually assume the base field to have characteristic $0$ (and algebraically closed?). My question is: how much can one prove over a positive ...
2
votes
0answers
59 views

Trigonometric multiple integral identity

How this alleged multiple integral identity can be proved? $$\int\limits_{-\infty }^{\infty }ds_{1}\int\limits_{-\infty}^{s_{1}}ds_{2}\cdots \int\limits_{-\infty }^{s_{2n-1}}ds_{2n}\;\cos { ...
18
votes
1answer
675 views

Interesting integral

Numerical evidence shows the validity of the following identity $$\int\limits_0^z\frac{xdx}{\sin{x}\sqrt{\sin^2{z}-\sin^2{x}}}=\frac{\pi}{4\sin{z}}\ln{\frac{1+\sin{z}}{1-\sin{z}}},\tag{1}$$ if $0< ...
2
votes
0answers
224 views

Inequality with CDF of order statistics

here is a problem I have been struggling with for a while now. This is for a paper I am working on. Any help would be appreciated! Here we go: Each bidder's valuation $\theta _{i},$ $i=1,...,N$, is ...
4
votes
2answers
575 views

Weak convergence of random measures

Let $\mu_n,n\in \mathbb N$ be a random probability measures and let $\mu$ be a deterministic probability measure on $\mathbb R$. That is to say, that the $\mu_n$ are measurable maps from a probability ...
3
votes
1answer
151 views

Is it possible to get an equation with two exponentials and a bessel function in closed form?

Is it possible to get the equation below into closed form? I have tried using integration tables but I haven't found anything that matches. Are there any other methods to achieve a closed form ...
3
votes
0answers
196 views

A difficult integral which the Risch algorithm shows is not elementary

For reasons which aren't conceptually related to the problem a few of my colleagues and I are in need of finding an expression for the following integral in terms of $a$ and $\delta$: ...
3
votes
2answers
320 views

Monte Carlo integration of Gaussian integrals

I was doing a physical problem, and then it comes to this Gaussian integral. The dimension of the integral is very large (dimension = 300~600), and it is difficult to find the maximum of the ...
0
votes
0answers
83 views

Integral over conditioning variable of a Gaussian

The marginal of a multivariate Gaussian can be computed in closed form, i.e., $p(x) = \int_y \mathcal{N}((x,y);\mu,\Sigma)\ dy$ is simple. But what I need is $L(x) = \int_y \mathcal{N}((x\mid y); ...
1
vote
0answers
347 views

convolution integral involving modified Bessel functions of the first kind

I'm stuck with this convolution integral ($z \geq 0$)... \begin{equation} f_{Z}(z)=\int^{\infty}_{-\infty}f_{1}(x)f_{2}(z-x)dx = \mbox{ } ??? \end{equation} which represents the pdf of the sum $Z = ...
0
votes
0answers
143 views

Asymptotic Expansion of Double integral

Crosspost from math.stackexchange. Have a look at the great answers there, even though they do not quite answer the question completely. Define $$G(\theta) = \int\limits_0^\infty \int\limits_0^{2\pi} ...
2
votes
0answers
228 views

A integral equation with Discrete to result by inverse problem

Problem I asked a question at Math.SE last year and later offered a bounty for it, but it remains unsolved even in the simplest case. So I finally decided to repost this case here, (I know the ...
1
vote
1answer
56 views

Expectation of logarithmic of a Laplace random varible

Say $Y$ is a random variable with Laplace distribution with zero mean and variance parameter $b$. I am trying to compute the expectation of $\ln(Y+\alpha)$ ($\alpha>0$), that is: ...
5
votes
0answers
301 views

Reference request : Grothendieck's topological space valued integral

As I am learning the different kind of Banach space valued integrals (Pettis, Bochner), I know that Grothendieck made a "mémoire" in his youth about this topic, but I don't know if it is available ...
1
vote
0answers
90 views

Convergence of solutions of the volterra integral equation with convergent kernels

Consider the following Volterra integral equation $$ g(t) = \int_0^t K_n(t,s)w_n(s) ds $$ where g(t) and K_n(t,s) are continuous and $K_n(t,s)\geq K_{n+1}(t,s)$ for all $t,s$. Moreover, $K_n(t,s)$ ...
1
vote
0answers
152 views

Reference : Special case of Banach-valued function integration by parts

Let $u \in L^p (0,T; L^1(\Omega))$ with $\partial_t u \in L^p(0,T; L^1(\Omega))$ and $v \in L^q (0,T; L^\infty(\Omega))$ with $\partial_t u \in L^q(0,T; L^\infty(\Omega))$ (with $1/p+1/q=1$ and $p \in ...
2
votes
0answers
221 views

Integral involving a Meijer-G function

I am having trouble with calculating the following integral: $$ \int_{0}^{\infty} \ln{(1 + \alpha x)\, G^{k,0}_{k,k}\left[e^{-x}\left|^{(a_k)}_{(b_k)} \right. \right]} \, dx, $$ where $\alpha > ...
0
votes
0answers
89 views

Do they have the same limit?

Suppose $a(\cdot)\in L^p$ and is symmetric and $b(\cdot)\in L^q$, where $1/p+2/q=2$, $p,q\ge 1$. Consider the quantity $Q_T=$ $$ ...
1
vote
0answers
111 views

Adelic integral factorization

In order to calculate Tamagawa numbers, I need to justify that for a nice (say Schwartz-Bruhat) function, the following identity holds : $$\int_{\mathbf{A}^2} f(x)dx = \int_{SL_2(\mathbf{A})/SL_2(K)} ...
0
votes
1answer
98 views

Can the conservative form of the advection equation be re-written by replacing the velocity term with an integral over all other points in space?

Suppose I have a 1D advection equation in conservation (divergence) form $\partial_t u(x,t) = -\partial_x [v(x)u(x,t)],$ where $u$ is a conserved quantity in space, and $v$ gives the velocity of the ...
0
votes
0answers
140 views

Integrate Faddeeva function

I came across this integration in my studies. $\int_{-\infty}^{\infty}|F((w_\textbf{_} - \hat{w_\textbf{_}})\tau) |^2 . d\tau$ It uses the Faddeeva function which is $F(z) = e^{-z^2}erfc(-iz)$. I ...
0
votes
0answers
449 views

How to solve definite integral involving exponential function

I am trying to get a closed form for the following definite integral: $$f(\theta)= \int_\frac{\pi}{2}^\pi \frac{1}{\sqrt{1-\alpha^2 \cos^2\theta}}\exp\left(C_2\cos\theta-C_1\sqrt{1-\alpha^2 ...
6
votes
1answer
173 views

Computing certain integrals over high-dimensional polyhedra

Let $\delta>0$ be a small real number and consider the $k$-dimensional region consisting of points for which $$\delta\leq x_1\leq x_2\leq\ldots \leq x_k$$ and $$x_1+\ldots+x_k\leq 1.$$ I am ...
3
votes
0answers
171 views

An upper bound for a average of a function in $L_{p}([0,1))$

Suppose that $ f $ is $ 1 $-periodic and that $ f \in {L^{p}}([0,1) $, where $ p > 1 $. Let $$ (D_{n})_{n \in \mathbb{N}_{0}} = \left( \left\{ I^{n}_{j},~ 1\leq j \leq 2^{n} \} \right\} \right)_{n ...
2
votes
0answers
152 views

Regularity of measures in the theorem of Riesz

There are two concurrent theories of measure/integration on a locally compact topological spaces: either as positive linear forms on the space of continuous functions with compact support, or as Borel ...
5
votes
2answers
199 views

Integrals involving trigonometric functions and polynomials

Can one describe all the real polynomials $P(x)$ such that the following integrals converge: $$ \int_0^{\infty} \sin(P(x))dx, \int_0^{\infty} \cos(P(x))dx ? $$ Among special cases are such ...
0
votes
0answers
165 views

Establishing an upper bound for a dyadic average of a function in $ {L^{p}}([0,1)) $

Suppose that $ f $ is $ 1 $-periodic and that $ f \in {L^{p}}([0,1)) $, where $ p > 1 $. Let $$ (D_{n})_{n \in \mathbb{N}_{0}} = \left( \left\{ I^{n}_{j} \stackrel{\text{df}}{=} \left[ ...
3
votes
0answers
108 views

motivic integration and jacobian ideal

When we consider the change of variables in motivic integration, we have a birational map $f:Y\rightarrow X$ with Y smooth and we have to consider two invariants the order of the Jacobian ideal of $X$ ...
0
votes
0answers
164 views

How to evaluate the following integral related to exponential distribution

I would like to evaluate the following integral related to the exponential distribution. Let $\delta>1$, and $0<p<1$ and $0<\epsilon<1/\delta$ be reals. We have that $$ ...